The left hand side of the last inequality, multiplied by \((-\tfrac{3}{2})\)  is called MTW-curvature or cost-curvature; it is denoted by \(\mathfrak {S}(X,Y)\) , see [1]};
if \(p=q\) , then \(\mathfrak {S}(X,Y)\)  coincides with the curvature \(\langle (X,Y)Y,X\rangle \) , see [1]}.
In particular, if the condition ( REF) holds, then the manifold has nonnegative sectional curvature.
